Presses have heretofore been made, for example as disclosed in the applicant's prior U.S. Pat. No. 4,669,346, with horizontally movable stock positioning apparatus that can be operated either manually or by numerically controlled mechanism, to move a sheet stock workpiece horizontally along first and second relatively transverse axes in the plane of the workpiece, to position different areas of the workpiece at a press work station. Presses with such horizontally movable carriages are adapted for use in performing piercing or blanking operations on the sheet stock workpiece. However, problems are encountered when it is desired to perform a forming operation on the workpiece in which different areas of the workpiece are drawn, embossed or otherwise deformed laterally out of the plane of the workpiece. Conventional presses have a vertically movable ram disposed above the work station. If the upper and lower press tools are arranged to perform a forming operation in which the workpiece is deformed downwardly from the plane of the workpiece, then the downwardly deformed portions on the workpiece can interfere with horizontal movement of the deformed areas of the workpiece past the lower press tools. The press can also be arranged so that the upper press tool presses the workpiece downwardly over the lower press tool to deform an area of the workpiece upwardly. In the prior presses with horizontally movable work positioning carriers known to the applicant, the carrier gripped and supported a large sheet adjacent its periphery at a fixed level and the sheet had to flex in areas inwardly of its gripped periphery in order to accommodate vertical movement of the area of the sheet engaged by the upper and lower press forming tools during a forming operation. In such machines the sheet could be laterally deformed only in areas spaced a substantial distance from the stock grippers. Moreover, flexing of the sheet limited the accuracy with which the forming operation could be performed.